**To construct a square-root spiral.**

**Objective: **To construct a square-root spiral.

**Material Required**: Coloured threads, adhesive, drawing pins, nails, geometry box, sketch pens, marker, and a piece of plywood.

**Method of Construction:**

- Take a piece of plywood with dimensions 30 cm × 30 cm.
- Taking 2 cm = 1 unit, draw a line segment AB of length one unit.
- Construct a perpendicular BX at the line segment AB using set squares (or compasses).
- From BX, cut off BC = 1 unit. Join AC.
- Using blue-coloured thread (of length equal to AC) and adhesive, fix the thread along AC.
- With AC as the base and using set squares (or compasses), draw CY perpendicular to AC.
- From CY, cut-off CD = 1 unit and join AD.

- Fix orange-coloured thread (of length equal to AD) along AD with adhesive.
- With AD as a base and using set squares (or compasses), draw DZ perpendicular to AD. From DZ, cut off DE = 1 unit and join AE.
- Fix green-coloured thread (of length equal to AE) along AE with adhesive [see Fig. 1].
- Repeat the above process for a sufficient number of times. This is called “a square root spiral”.

**Demonstration**

****From the figure, AC^{2 }= AB^{2} + BC^{2} = 1^{2} + 1^{2} = √2 or AC = 2.

AD^{2} = AC^{2} + CD^{2} = 2 + 1 = 3 or AD = √3 .

Similarly, we get the other lengths AE, AF, AG, ... as √4 or 2, √5, 6...

**Observation**

****On actual measurementAC = ..... , AD = ...... , AE =...... , AF =....... , A

G = ......√2 = AC = ............... (approx.),√3 = AD = ............... (approx.),

√4 = AE = ............... (approx.),√5 = AF = ............... (approx.)****

**Application**

****Through this activity, the existence of irrational numbers can be illustrated.

**To construct a square-root spiral.**

**Objective: **To represent some irrational numbers on the number line.

**Material Required**: Two cuboidal wooden strips, thread, nails, a hammer, two photocopies of a scale, a screw with nut, glue, and a cutter.

**Method of Construction:**

- Make a straight slit on the top of one of the wooden strips. Fix another wooden strip on the slit perpendicular to the former strip with a screw at the bottom so that it can move freely along the slit [see Fig.1].
- Paste one photocopy of the scale on each of these two strips as shown in Fig. 1.
- Fix nails at a distance of 1 unit each, starting from 0, on both strips as shown in the figure.
- Tie a thread at the nail at 0 on the horizontal strip.

**Demonstration**

Take 1 unit on the horizontal scale and fix the perpendicular wooden strip at 1 by the screw at the bottom.

Tie the other end of the thread to unit ‘1’ on the perpendicular strip.

Remove the thread from unit ‘1’ on the perpendicular strip and place it on the horizontal strip to represent √2 on the horizontal strip [see Fig. 1].

Similarly, to represent √3, fix the perpendicular wooden strip at √2 and repeat the process as above. To represent √a, a > 1, fix the perpendicular scale at √a – 1 and proceed as above to get √a

**Observation**

On actual measurement:

a – 1 = ........... √a = ...........

**Application**

The activity may help in representing some irrational numbers such as √2, √3, √4, √5, √6, √7, .... on the number line.

**Note**

You may also find √a such as √13 by fixing the perpendicular strip at √3 on the horizontal strip and tying the other end of the thread at 2 on the vertical strip.